Taras Goy and Mark Shattuck

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 26, 2020, Number 3, Pages 203–217

DOI: 10.7546/nntdm.2020.26.3.203-217

**Full paper (PDF, 232 Kb)**

## Details

### Authors and affiliations

Taras Goy

*Faculty of Mathematics and Computer Science
Vasyl Stefanyk Precarpathian National University
57 Shevchenko St., 76018 Ivano-Frankivsk, Ukraine
*

Mark Shattuck

*Department of Mathematics, University of Tennessee
37996 Knoxville, TN, USA*

### Abstract

In this paper, we evaluate determinants of several families of Hessenberg matrices having Fibonacci numbers as their nonzero entries. By the generalized Trudi formula, these determinant identities may be written equivalently as formulas for the Lucas numbers in terms of the Fibonacci. We provide both algebraic and combinatorial proofs of our determinant results. The former makes use of expansion along columns and induction, while the latter draws upon the definition of the determinant as a signed sum over the symmetric group and uses parity-changing involutions.

### Keywords

- Hessenberg matrix
- Fibonacci number
- Determinant
- Trudi formula
- Lucas number

### 2010 Mathematics Subject Classification

- 05A19
- 11B39
- 15B05

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## Cite this paper

Goy, T. & Shattuck, M. (2020). Fibonacci–Lucas identities and the generalized Trudi formula. *Notes on Number Theory and Discrete Mathematics*, 26 (3), 203-217, DOI: 10.7546/nntdm.2020.26.3.203-217.